Although Alfred Haar first mentioned the term wavelet in a 1909 thesis, the idea of wavelet analysis did not receive much attention until the late 1970s. Since then, wavelet analysis has been studied thoroughly and applied successfully in many areas. Current wavelet analysis is thought by some to be no more than the recasting and unifying of existing theories and techniques. Nevertheless, the breadth of applications for wavelet analysis is more expansive than ever was anticipated.
The development of wavelet analysis originally was motivated by the desire to overcome the drawbacks of traditional Fourier analysis and short-time Fourier transform processes. The Fourier transform characterizes the frequency behaviors of a signal, but not how the frequencies change over time. The Short-time Fourier transform, or windowed Fourier transform, simultaneously characterizes a signal in time and frequency. However, problems may be encountered because the signal time and frequency resolutions are fixed once the type of window is selected. However, signals encountered in nature always have a long time period at low frequency and a short time period at high frequency. This suggests that the window should have high time resolution at high frequency.
To understand the fundamentals of wavelet analysis, consider the following artificial example. FIG. 1 shows a signal s(t) that consists of two truncated sine waveforms. While the first waveform spans 0 to 1 second, the second waveform spans 1 to 1.5 seconds. In other words, the frequency of s(t) is 1 Hz for 0.ltoreq.t&lt;1 and 2 Hz for 1.ltoreq.t&lt;1.5.
When describing frequency behavior, s(t) is traditionally compared with a group of harmonically related complex sinusoidal functions, such as exp{j2 .pi.kt/T}. Here, the term harmonically related complex sinusoidal functions refers to the sets of periodic sinusoidal functions with fundamental frequencies that are all multiples of a single positive frequency 2 .pi./T. The comparison process is accomplished with the following correlation (or inner product) operation: ##EQU1## where a.sub.k is the Fourier coefficient, and * denotes a complex conjugate.
The magnitude of a.sub.k indicates the degree of similarity between the signal s(t) and the elementary function exp{j2 .pi.kt/T}. If this quantity is large, it indicates a high degree of correlation between s(t) and exp {j2 .pi.kt/T}. If this quantity is almost 0, it indicates a low degree of correlation between s(t) and exp{j2 .pi.kt/T}. Therefore, a.sub.k can be considered as the measure of similarity between the signal s(t) and each of the individual complex sinusoidal functions exp{j2 .pi.kt/T}. Because exp{j2 .pi.kt/T} represents a distinct frequency 2 .pi.k/T (a frequency tick mark), the Fourier coefficient a.sub.k indicates the amount of signal present at the frequency 2 .pi.k/T.
In FIG. 1, Sum of Two Truncated Sine Waveforms, s(t) consists of two truncated sine waveforms. The inner product of such truncated signal and pure sine waveforms, which extends from minus infinity to plus infinity, never vanishes, that is, a.sub.k is not zero for all k. However, the dominant a.sub.k, with the largest magnitude, is that which corresponds to 1 and 2 Hz elementary functions. This indicates that the primary components of s(t) are 1 and 2 Hz signals. However, it is unclear, based on a.sub.k alone, when the 1 Hz or the 2 Hz components exist in time.
There are many ways of building the frequency tick marks to measure the frequency behavior of a signal. By using complex sinusoidal functions, not only can the signals be analyzed, but the original signal can be reconstructed with the Fourier coefficient a.sub.k. For example, s(t) can be written in terms of the sum of complex sinusoidal functions, according to the following formula, traditionally known as the Fourier expansion. ##EQU2## where a.sub.k is the Fourier coefficient and 2 .pi.k/T is the frequency tick mark. In this equation, because a.sub.k is not zero for all k, an infinite number of complex sinusoidal functions must be used in (3-2) to restore s(t) in FIG. 1, Sum of Two Truncated Sine Waveforms.
Wavelet Analysis
Looking at s(t) more closely, to determine the frequency contents of s(t), information is only needed regarding one cycle, such as the time span of one cycle. With this information, the frequency can be computed with the following formula: ##EQU3##
According to this equation, the higher the frequency, the shorter the time span. Therefore, instead of using infinitely long complex sinusoidal functions, it is only necessary to use one cycle of a sinusoidal waveform, or a wavelet, to measure s(t). The wavelet .psi.(t) used to measure s(t) is one cycle of sinusoidal waveform, as shown in FIG. 2.
Because .psi.(t) spans 1 second, consider the frequency of .psi.(t) to be 1 Hz. As in the case of Fourier analysis, the comparison process can be achieved with the following correlation (or inner product) operation: ##EQU4## where W.sub.m,n denotes the wavelet transform coefficients and .psi..sub.m,n (t) are the elementary functions of the wavelet transform.
However, the structure of the elementary functions .psi..sub.m,n (t) differs from the Fourier transformations, which are the dilated and shifted versions of .psi.(t), that is, EQU .psi..sub.m,n (t)=2.sup.m/2 .psi.(2.sup.m (t-n2.sup.-m)) (3-4)
where m and n are integers.
By increasing n, .psi..sub.m,n (t) is shifted forward in time. By increasing m, the time duration is compressed which thereby increases the center frequency and frequency bandwidth of .psi.(t). For more information on this, please see Qian and Chen, Joint Time Frequency Analysis, Prentice-Hall 1996. The parameter m can be considered as the scale factor and 2.sup.-m as the sampling step. Therefore, the shorter the time duration, the smaller the time sampling step, and vice versa. Assuming the center frequency of .psi.(t) is coo, then the center frequency of .psi..sub.m,n (t) would be 2.sup.m .omega..sub.0. Consequently, the scale factor m can be systematically adjusted to achieve different frequency tick marks to measure the signal frequency contents. That is, as the scale factor m increases, the center frequency and bandwidth of the wavelet increases 2.sup.m.
FIG. 3 depicts the wavelet transform procedure. First, let m=n=0, that is, align .psi.(t) and s(t) at t=0. Then, as in equation (3-4), compare .psi.(t) with s(t) for 0.ltoreq.t&lt;1. W.sub.0,0 =1 is obtained. Shift .psi.(t) to the next second, that is, let n=1, and compare it with s(t) for 1.ltoreq.t&lt;2. W.sub.0,1 =0 is obtained.
Next, compress Mf(t) into 0 to 0.5 seconds, that is, let m=1, and repeat the previous operations with the time-shift step 0.5. The following results are obtained, also displayed in the shaded table of FIG. 3, Wavelet Analysis, from this process: EQU W.sub.1,0 =0 W.sub.1,1 =0 W.sub.1,2 =1 W.sub.1,3 =0
One can continue to compress .psi.(t) by increasing the scale factor m and reducing the time-shift step 2.sup.-m to test s(t). This procedure is called the wavelet transform. .psi.(t) is called the mother wavelet because the different wavelets used to measure s(t) are the dilated and shifted versions of this wavelet. The results of each comparison, W.sub.m,n, are named wavelet coefficients. The index m and n are the scale and time indicators, respectively, which describe the signal behavior in the joint time-scale domain (As shown in FIG. 5, Wavelet Transform Sampling Grid, the scale can be easily converted into frequency. Hence, W.sub.m,n also can be considered the signal representation in the joint time and frequency domain). In this example, by checking the wavelet coefficients, it is known that for 0.ltoreq.t&lt;1 the frequency of s(t) is 1 Hz and for 1.ltoreq.t&lt;1.5 the frequency of s(t) is 2 Hz. Unlike Fourier analysis, the wavelet transform not only indicates what frequencies the signal s(t) contains, but also when these frequencies occur. Moreover, the wavelet coefficients W.sub.m,n of a real-valued signal s(t) are always real as long as real-valued .psi.(t) is chosen. Compared to the Fourier expansion, fewer wavelet functions can usually be used to represent the signal s(t). In this example, s(t) can be completely represented by two terms, whereas an infinite number of complex sinusoidal functions would be needed in the case of Fourier expansion.
Wavelet Analysis vs. Fourier Analysis
The short-time Fourier transform can be applied to characterize a signal in both the time and frequency domains simultaneously. However, wavelet analysis can also be used to perform the same function because of its similarity to the short-time Fourier transform. Both are computed by the correlation (or inner product) operation, but the main difference lies in how the elementary functions are built.
For the short-time Fourier transform, the elementary functions used to test the signal are time-shifted, frequency-modulated single window functions, all with some envelope. Because this modulation does not change the time or frequency resolutions (Qian and Chen 1996), the time and frequency resolutions of the elementary functions employed in short-time Fourier transform are constant. FIG. 4 illustrates the sampling grid for the short-time Fourier transform.
For the wavelet transform, increasing the scale parameter m reduces the width of the wavelets. The time resolution of the wavelets improves and the frequency resolution becomes worse as m becomes larger. Because of this, wavelet analysis has good time resolution at high frequencies and good frequency resolution at low frequencies.
FIG. 5 illustrates the sampling grid for the wavelet transform. Suppose that the center frequency and bandwidth of the mother wavelet .psi.(t) are .omega..sub.0 and .DELTA..sub..omega., respectively. Then, the center frequency and bandwidth of .psi.(2.sup.m t) are 2.sup.m .omega..sub.0 and 2.sup.m .DELTA..sub..omega.. Although the time and frequency resolutions change at different scales m, the ratio between bandwidth and center frequency remains constant. Therefore, wavelet analysis is also called constant Q analysis, where Q=center frequency/bandwidth.
The wavelet transform is closely related to both conventional Fourier transform and short-time Fourier transform. As shown in FIG. 6, Comparison of Transform Processes, all these transform processes employ the same mathematical tool, the correlation operation or inner product, to compare the signal s(t) to the elementary function b.sub..alpha. (t). The difference lies in the structure of the elementary functions {e.sub..alpha. (t)}. In some cases, wavelet analysis is more natural because the signals always have a long time cycle at low frequency and a short time cycle at high frequency.
Background on the design of two-channel perfect reconstruction filter banks and the types of filter banks used with wavelet analysis is deemed appropriate.
Digital Filter Banks
The wavelet transform can be implemented with specific types of digital filter banks known as two-channel perfect reconstruction filter banks. The following describes the basics of two-channel perfect reconstruction filter banks and the types of digital filter banks used with wavelet analysis. The following describes the design of two-channel perfect reconstruction filter banks and defines the types of filter banks used with wavelet analysis.
Two-Channel Perfect Reconstruction Filter Banks
Two-channel perfect reconstruction (PR) filter banks were recognized as useful in signal processing for a long time, particularly after their close relationship with wavelet transform was discovered. Since then, it has become a common technique for computing wavelet transform.
FIG. 7 illustrates a typical two-channel filter bank system. The signal X(z) is first filtered by a filter bank constituted by G.sub.0 (z) and G.sub.1 (z).
It is noted that, for a finite impulse response (FIR) digital filter g[n], its z-transform is defined as ##EQU5## where N denotes the filter order. Consequently, the filter length is equal to N+1. Clearly, .omega.=0 is equivalent to z=1. .omega.=.pi. is equivalent to z=-1. That is, G(O) and G(.pi.) in the frequency domain correspond to G(1) and G(-1) in the z-domain.
Then, outputs of G.sub.0 (z) and G.sub.1 (z) are downsampled by 2 to obtain Y.sub.0 (z) and Y.sub.1 (z). After some processing, the modified signals are upsampled and filtered by another filter bank constructed by H.sub.0 (z) and H.sub.1 (z). If no processing takes place between the two filter banks, that is, Y.sub.0 (z) and Y.sub.1 (z) are not altered, the sum of outputs of H.sub.0 (z) and H.sub.1 (z) is identical to the original signal X(z), except for the time delay. Such a system is commonly referred to as two-channel PR filter banks. G.sub.0 (z) and G.sub.1 (z) form an analysis filter bank, whereas H.sub.0 (z) and H.sub.1 (z) form a synthesis filter bank.
It is noted that G(z) and H(z) can be interchanged. For instance, H.sub.0 (z) and H.sub.1 (z) can be used for analysis and G.sub.0 (z) and G.sub.1 (z) can be used for synthesis. H.sub.0 (z) and H.sub.1 (z) are usually considered as the dual of G.sub.0 (z) and G.sub.1 (z), and vice versa. Traditionally, G.sub.0 (z) and H.sub.0 (z) are lowpass filters, while G.sub.1 (z) and H.sub.1 (z) are highpass filters, where the subscripts 0 and 1 represent lowpass and highpass filters, respectively. Because the two-channel PR filter banks process Y.sub.0 (z) and Y.sub.1 (z) at half the sampling rate of the original signal X(z), they attract many signal processing applications.
If the convention illustrated in FIG. 8 is assumed, then the relationship between two-channel PR filter banks and wavelet transform can be illustrated by FIG. 9.
It is proven (Qian and Chen 1996) that under certain conditions, two-channel PR filter banks are related to wavelet transform in two ways:
The impulse response of the lowpass filters converges to the scaling function .phi.(t). Once .phi.(t) is obtained, the mother wavelet function .psi.(t) can be computed by highpass .phi.(t), as shown in FIG. 9.
The outputs of each of the highpass filters are approximations of the wavelet transform. The wavelet transform can be accomplished with a tree of two-channel PR filter banks. The selection of a desirable mother wavelet becomes the design of two-channel PR filter banks.
FIG. 10 illustrates the relationship of filter banks and wavelet transform coefficients.
The following sections describe the design fundamentals for two types of two-channel PR filter banks, biorthogonal and orthogonal.
Biorthogonal Filter Banks
Referring back to FIG. 8, Two-Channel Filter Bank, the output of the low-channel can be defined as ##EQU6##
Similarly, the output of the up-channel can be defined as ##EQU7##
Add them together to obtain ##EQU8##
One term involves X(z) while the other involves X(-z). For perfect reconstruction, the term with X(-z), traditionally called the alias term, must be zero. To achieve this, the following equation is used, EQU H.sub.0 (z)G.sub.0 (-z)+H.sub.1 (z)G.sub.1 (-z)=0, (3-9)
which is accomplished by letting ##STR1##
The relationship in equation (3-10) implies that h.sub.0 [n] can be obtained by alternating the sign of g.sub.1 [n], that is, EQU h.sub.0 [n]=(-1).sup.n g.sub.1 [n]. (3-11)
Similarly, EQU h .sub.1 [n]=(-1).sup.n+1 g.sub.0 [n]. (3-12)
Therefore, g.sub.1 [n] and h.sub.1 [n] are the highpass filters if g.sub.0 [n] and h.sub.0 [n] are the lowpass filters. For perfect reconstruction, the first term in equation (3-8), called the distortion term, should be a constant or a pure time delay. For example, EQU .sub.0 (z)G.sub.0 (z)+H.sub.1 (z)G.sub.1 (z)=2z.sup.-1, (3-13)
where l denotes a time delay.
If both equations (3-9) and (3-13) are satisfied, the output of the two-channel filter bank in FIG. 7, Two-Channel Filter Bank, is a delayed version of the input signal, that is, EQU X(z)=z.sup.-1 X(z). (3-14)
However, there remains a problem computing G.sub.0 (z) and G.sub.1 (z) [or H.sub.0 (z) and H.sub.0 (z)]. Once G.sub.0 (z) and G.sub.1 (z) are determined, the rest of the filters can be found with equation (3-10).
Equation (3-10) can be written as, EQU G.sub.1 (z)=H.sub.0 (-z) and H.sub.1 (z)=-G.sub.0 (-z).
Substituting it into equation (3-13) yields, ##STR2## where P.sub.0 (z) denotes the product of two lowpass filters, G.sub.0 (z) and H.sub.0 (z), that is, EQU .sub.0 (z)=G.sub.0 (z)H.sub.0 (z). (3-16)
Equation (3-15) indicates that all odd terms of the product of two lowpass filters, G.sub.0 (z) and H.sub.0 (z), must be zero except for order l, where l must be odd. But, even order terms are arbitrary. These observations can be summarized by the following formula: ##EQU9## This reduces the design of two-channel PR filter banks to two steps: 1. Design a filter P.sub.0 (z) satisfying equation (3-17).
2. Factorize P.sub.0 (z) into G.sub.0 (z) and H.sub.0 (z). Then use equation (3-10) to compute G.sub.1 (z) and H.sub.1 (z).
Two types of filters are frequently used for P.sub.0 (z):
an equiripple halfband filter (Vaidyanathan and Nguyen 1987) PA1 a maximum flat filter PA1 low-low PA1 low-high PA1 high-low PA1 high-high
In the first filter, the halfband refers to a filter in which .omega..sub.s +.omega..sub.p =.pi., where .omega..sub.s and .omega..sub.p denote the passband and stopband frequencies, respectively, as in FIG. 11, Halfband Filter.
The second filter is the maximum flat filter with a form according to the following formula: EQU P.sub.0 (z)=(1+z.sup.-1).sup.2p Q(z), (3-18)
which has 2 p zeros at z=-1 or .omega.=.pi.. If the order of the polynomial Q(z) is limited to 2 p-2, then Q(z) is unique.
The maximum flat filter here differs from the Butterworth filter. The low-frequency asymptote of the Butterworth filter is a constant, while the maximum flat filter is not.
In all cases, the product of lowpass filter P.sub.0 (z) is a type I filter, that is, EQU p.sub.0 [n]=p.sub.0 [N-n]N even, (3-19)
where N denotes the filter order. Consequently, the number of coefficients p.sub.0 [n] is odd, N+1.
FIG. 12 plots the zeros distribution of a maximum flat filter P.sub.0 (z) for p=3.
There are six zeros at .omega.=.pi.. In this case, the order of the unique polynomial Q(z) is four, which contributes another four zeros (not on the unit circle). If three zeros at .omega.=.pi. go to G.sub.0 (z) according to the formula, EQU G.sub.0 (z)=(1+z.sup.-1).sup.3, (3-20)
and the rest of the zeros go to H.sub.0 (z), B-spline filter banks are obtained. The coefficients of g.sub.0 [n] and g.sub.1 [n] and the corresponding scaling function and mother wavelet are plotted in FIG. 13, B-spline Filter Bank. Both the scaling function and mother wavelet generated by g.sub.0 [n] and g.sub.1 [n] are smooth.
FIG. 14 depicts the dual filter bank, h.sub.0 [n] and h.sub.1 [n], and corresponding scaling function and mother wavelet. One also can use h.sub.0 [n] and h.sub.1 [n] for analysis. In FIG. 14, the tree filter banks constituted by h.sub.0 [n] and h.sub.1 [n] do not converge.
It is noted that two-channel PR filter banks do not necessarily correspond to the wavelet transform. The wavelet transformations are special cases of two-channel PR filter banks. The conditions of two-channel PR filter banks are more moderate than those for the wavelet transform.
Finally, the analysis filter banks and synthesis filter banks presented in this section are orthogonal to each other. That is ##EQU10##
The filters banks that satisfy equation (3-12) are traditionally called biorthogonal filter banks. In addition to equation (3-21), if the analysis filter banks also satisfy the following equations ##EQU11## the resulting filter banks are called orthogonal filter banks. Orthogonal filter banks are special cases of biorthogonal filter banks.
Orthogonal Filter Banks
As shown in the preceding section, once P.sub.0 (z) is determined, the product of two lowpass filters, P.sub.0 (z) must be factorized into G.sub.0 (z) and H.sub.0 (z). Evidently, the combinations of zeros are not unique. Different combinations lead to different filter banks. Sometimes G.sub.0 (z) and G.sub.1 (z) work well, but H.sub.0 (z) and H.sub.1 (z) might not (see FIG. 13, B-spline Filter Bank, and FIG. 14, Dual B-spline Filter Bank). One way to make this process easier is to limit the selections into a subset. The most effective approach is to require G.sub.0 (z) and G.sub.1 (z), and thereby H.sub.0 (z) and H.sub.1 (z), to be orthogonal, as described by equation (3-22).
These constraints reduce the filter banks design to one filter design. Once G.sub.0 (z) is selected, one can easily find all other filters. The constraints imposed by equation (3-22) not only guarantee that both filter banks have the same performance, but also provide other advantages. For example, many applications demonstrate that the lack of orthogonality complicates quantization and bit allocation between bands, eliminating the conservation of energy. To achieve equation (3-22), let EQU G.sub.1 (z)=-z.sup.-N G0(-z.sup.-1), (3-23)
which implies that g.sub.1 [n] is alternating flip of g.sub.0 [n], that is, EQU (g.sub.1 [0],g.sub.1 [1],g.sub.1 [2], . . .)=(g.sub.0 [N],-g.sub.0 [N-1], g.sub.0 [N-2], . . .). (3-24)
Equation (3-23) implies that for orthogonal wavelets and filter banks, EQU H.sub.0 (z)=z.sup.-N G.sub.0 (z.sup.-1), (3-25)
where the relation in equation (3-10) is used. Consequently, equation (3-16) can be written as, EQU P.sub.0 (z)=z.sup.-N G.sub.0 (z)G.sub.0 (z.sup.-1). (3-26)
If, EQU G.sub.0 (z)G.sub.0 (z.sup.-1)=P(z), (3-27)
then, ##EQU12## which implies that P(z) is non-negative.
Similar to biorthogonal cases, the selection of P.sub.0 (z) in orthogonal cases is dominated by maximum flat and equiripple halfband filters. However, because of constraints imposed by equation (3-28), P.sub.0 (z) must be the time-shifted non-negative function P(z). While the maximum flat filter in equation (3-18) ensures this requirement, special care must be taken when P.sub.0 (z) is an equiripple halfband filter.
FIG. 15 plots the third order Daubechies filter banks and wavelets. It is derived from the same maximum flat filter as that depicted in FIG. 12, Zeros Distribution for (1-z.sup.-1).sup.6 Q(z), but in this case, G.sub.0 (z) contains three zeros at .omega.=.pi. as well as all zeros inside of the unit circle, therefore possessing minimum phase. Because of the orthogonality, its dual filter bank has the same convergence property. Compared to the B-spline cases in FIG. 13, B-spline Filter Bank, and FIG. 14, Dual B-spline Filter Bank, the third order Daubechies wavelet and scaling function are less smooth than that of G.sub.0 (z) and G.sub.1 (z) (see FIG. 13, B-spline Filter Bank), but much smoother than that of H.sub.0 (z) and H.sub.1 (z) (see FIG. 14, Dual B-spline Filter Bank).
Two-Dimensional Signal Processing
The preceding sections introduced two-channel PR filter banks for one-dimensional (1D) signal processing. In fact, two-channel PR filter banks also can be used for two-dimensional (2D) signals as shown in FIG. 16. In this case, the rows are processed first and then columns. Consequently, one 2D array splits to the following four 2D sub-arrays:
Each sub-arrays is a quarter size of the original 2D signal.
FIG. 29, 2D Image Decomposition, illustrates 2D image decomposition by two-channel PR filter banks. In this case, the original 128-by-128 2D array is decomposed into four 64-by-64 sub-arrays, but the total size of the four sub-arrays is the same as the original 2D array. For example, the total number of elements in the four sub-arrays is 16,384, which equals 128 times 128. However, if the filters are selected properly, sub-arrays can be made such that the majority elements are small enough to be neglected. Consequently, a fraction of the entire wavelet transform coefficients can be used to recover the original image and thereby achieve data compression. In this example, the largest 25% wavelet transform coefficients are used to rebuild the original image. Among them, the majority (93.22%) are from the low-low sub-array. The remaining three sub-arrays contain limited information. If the wavelet transform is repeated to the low-low sub-array, the compression rate can be reduced further.
Selection of the Mother Wavelet
Although wavelet analysis possesses many attractive features, its numerical implementation is not as straightforward as its counterparts, such as the conventional Fourier transform and short-time Fourier transform. The difficulty arises from the following two aspects.
In order to reconstruct the original signal, the selection of the mother wavelet .psi.(t) is not arbitrary. Although any function can be used in (3-3), the original signal sometimes cannot be restored based on the resulting wavelet coefficients W.sub.m,n. .psi.(t) is a valid or qualified wavelet only if the original signal can be reconstructed from its corresponding wavelet coefficients. The selection of the qualified wavelet is subject to certain restrictions. On the other hand, it is not unique. Unlike the case of conventional Fourier transform, in which the basis functions must be complex sinusoidal functions, one can select from an infinite number of mother wavelet functions. Therefore, the biggest issue of applying wavelet analysis is how to choose a desired mother wavelet .psi.(t). It is generally agreed that the success of the application of wavelet transform hinges on the selection of a proper wavelet function.
Because the scale factor m could go from negative infinity to positive infinity, it is impossible to make the time index of the wavelet function, 2.sup.m (t-n2.sup.-m), an integer number simply by digitizing t as i.DELTA..sub.t, where .DELTA..sub.t denotes the time sampling interval. This problem prohibits us from using digital computers to evaluate wavelet transform.
Fortunately researchers discovered a relationship between the wavelet transform and the perfect reconstruction filter bank, a form of digital filter banks. The wavelet transform can be implemented with specific types of digital filter banks known as two-channel perfect reconstruction filter banks.
Currently there is a very limited number of available mother wavelets used for wavelet analysis. Applicant is aware that MatLab program from MathWorks which was released in July 1996?, includes a graphical method for choosing a mother wavelet. However, the Matlab program only allows the user to choose from pre-configured wavelets, and does not allow a user to design a new mother wavelet which is optimum for the user's application.
Therefore, a system and method is desired for designing a mother wavelet for use in wavelet analysis. A system and method is also desired which provides an intuitive graphical interface for designing the mother wavelet.